I'm asking about an area (Hamiltonian mechanics) that I don't know at all well; thus, I keep the question somewhat vague.
In differential geometry, there are a number of results saying that geodesics on a manifold with curvature $C$ bounded above and below by $0 \leq \delta \leq C \leq \Delta \leq \infty$ will have behaviour that is in some sense bounded between the behaviour of geodesics on the sphere with curvature $\delta$ and the behaviour of geodesics on the sphere with curvature $\Delta$. To give the example that I am most interested, Rauch's comparison theorem bounds the extent to which geodesics on a positively/negatively curved space tend to bend towards/away from each other over a moderate amount of time.
I am looking for analogous results on solutions to Hamiltonian mechanics problems with (strongly) convex potential functions. That is, I am looking for results saying that, if a potential is strongly convex with parameter $m$, then the solutions to the associated Hamiltonian mechanics problem can be compared to solutions of the problem with potential $U(x) = m |x|^{2}$. I am most interested in results comparing the rate at which solutions tend to bend towards/away from each other over a moderate amount of time.
I have not tried seriously to prove this sort of bound, beyond noting that I seem to get something like this for the explicit potentials that I've tried (e.g. $U(x) = \langle Ax, Ax \rangle$ for some positive-definite matrix $A$, as well as perturbations of these potentials) and having been told that strict convexity can play a similar role to positive curvature.
Thanks for any pointers!
